We study the relationship between the Banach-Mazur distance and the modified Banach-Mazur distance. We sharpen the Szarek theorem on the construction of invariant norms with specific properties and construct extremally distant normed spaces with norms invariant under groups of automorphisms of small cardinality. We prove that the trivial upper bound for the Banach-Mazur distance obtained in terms of the modified Banach-Mazur distance is order-sharp.